I need to screw down securely a sheet of plywood onto a fiberglass panel. I thought I would just make it easy be getting some self tapping wood screws. These worked great going through the plywood, but when they hit the fiberglass it may as well been steel.

So what are the best type of screws to use for fiberglass? I could use lag bolts in this application also.

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jobless, houseless, clueless, living on a boat and cruising around somewhere

Of course depends on service. But, drill and tap the fibreglass for a fine threaded screw/bolt to meet your service needs and set it with epoxy. If you need to remove it just heat it a bit (soldering iron) and back it out. The threads will remain for use as well. This method holds better than self tappers and does not crack the glass then allowing an easily pulled screw.

And here I thought this question would get a bunch of answers. My guess it that people are mainly using "whatever" with no real understanding of it being the right screw.

From internet searching it appears sheet metal screws are the right ones for fiberglass. So I guess fiberglass is more like metal than wood. But there is conflicting stuff out there and maybe the screw type doesn't really matter.

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jobless, houseless, clueless, living on a boat and cruising around somewhere

Yes, sheet metal screws. Personally I pilot with a size slightly bigger than the shank. Also, I like to lube the screws up with a little dish soap to make things go easier. Less likely to breaky breaky.

It would depend on whether you are trying to screw into a frp sheet, i.e. 1/16" thick ? Or something heavier, perhaps 3/8" thick.

In either case if the plywood is going to remain attached, I think I'd rather use Liquid Nails or a similar "construction adhesive".

Stainless steel or other corrosion resistant deck screws (for home decking) are very similar to drywall screws. Except of course, they are designed to be corrosion resistant. Just don't buy the really good ones, which have break-away heads designed so they bury flush in the decking.

I need to screw down securely a sheet of plywood onto a fiberglass panel. I thought I would just make it easy be getting some self tapping wood screws. These worked great going through the plywood, but when they hit the fiberglass it may as well been steel.

So what are the best type of screws to use for fiberglass? I could use lag bolts in this application also.

If you were to remove an existing screw on your boat you would probably find it to be a stainless steel sheet metal screw (sheet metal screws have threads from the tip to the head, wood screws have threads only part of the way from the tip to the head).

To install screws into fiberglass, you must first drill a pilot hole the size of the solid part of the screw or slightly larger. If the screws are really hard to drive, you need a bigger pilot hole.

To keep the gelcoat from cracking, use a countersink bit to chamfer the holes before inserting the screws.

The pilot holes in the plywood should be big enough for the screw shanks to slip through. That way, when you screw the screws into the fiberglass, the plywood will be pulled down tight to it.

As for folks using "whatever", I hope not. I think that most of us who do at least some of out own maintenance have already figured this out.

I need to screw down securely a sheet of plywood onto a fiberglass panel. I thought I would just make it easy be getting some self tapping wood screws. These worked great going through the plywood, but when they hit the fiberglass it may as well been steel.

So what are the best type of screws to use for fiberglass? I could use lag bolts in this application also.

You left out the most important part; the application. Are you just trying to hold the plywood in position on the floor/deck, or is it suspended from the overhead? Are the forces working against it in shear or tension? What thickness fiberglass are you fastening to, and what is the thickness and type (species) of the plywood you are fastening?

All these things can and should have bearing on the fastener selection, all the suggestions so far seem sound to me (except maybe the drywall screw one, I'm still finding drywall screws [and their remnants] that I'm sure somebody put in, intending them to be 'temporary'.)

The most important things are the proper size pilot hole, lubrication or sealant (they can usually be considered to perform the same purpose.) You will twist off any stainless screw by trying to screw it into a too-small pilot hole in any but the thinnest fiberglass panel.

For general purposes I most often use a coarser wood screw type thread, with a star drive or torx head if I can find and afford them. With the proper size pilot hole you can actually just screw a machine screw directly into the panel. Lubricate the threads with epoxy and it's probably stronger than sheet metal or wood screw threads, but I haven't tested it, as the sheet metal or wood screws are adequate and more convenient.

Best screws, none.
My question to you is why would you put a screw into fiberglass?
I refit a large portion of my interior wood paneling, where I was placing 1/2 marine ply with teak laminate directly onto cored fiberglass. I did not use a single screw. I bonded all of the ply with adhesives.
Placing a screw into fiberglass is a poor way to attach a wood panel for many reasons, namely it is a poor bond over time. Fiberglass shatters when you screw into it, and creates a hole that does not seal. On the exterior you are going to get water migration, then leaks.
All of the best boat builders stopped using screws in fiberglass a long time ago, and changed to adhesives.